There is an infinite number of rational solutions
$$a=\left(\frac{n+1}{n}\right)^n,\;\;b=\left(\frac{n+1}{n}\right)^{n+1},\;\;n\in\mathbb{Z},\;\;0\neq n\neq -1.$$
For a proof that these are *all* the rational solutions, see <A HREF="http://mathforum.org/library/drmath/view/67256.html">this posting.</A>